# Tagged Questions

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### Constraint optimization with lagrangian

I am having trouble regarding the general steps one needs to take in order to solve an constraint optimization using Lagrangian. More specifically, I want to maximize objective equation $f(x,y,z,w)$ ...
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### Linear constraint in convex optimization

Is it true that the solution to a linearly constrained convex minimization problem can only be placed on the boundary of the constraint set, for any nonlinear convex objective, e.g. $$\min_x f(x)$$ ...
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### Convexity of a rational function

I am attempting to (dis)prove that the function $$\frac{4x+3y+2}{x^2+xy+2x+y}$$ is convex for $x,y>0$. Attempting to differentiate the function does not seem like a good idea (or am I making a ...
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### Finding the maximum/minimum of a homogeneous function on $R^n$

Suppose that $f:R^n\to R$ is homogeneous. Also, suppose that the $argmin_xf(x)$ is non-empty. Is it true that if there exist $x^*\in R^n$ such that $f(x^*)=0$, then $x^*=argmin_xf(x)$?
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### Partial derivative on convex set

If we have a function $f:U \rightarrow R$ ($U \subset R^n$) which is partially differentiable on a convex set U with $\frac{\delta f}{\delta x_1} = 0$ for all $x \in U$. How can we prove that $f$ ...
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### Global Min-Max Optimization

When is $$\min_X \max_Y f(X,Y)$$ globally solvable? (i.e. we can find global solution for the optimization problem?) I am not looking for reformulations. Is it only when ...
Given $x \geq y > 0$ and an integer $n$. We want to minimize the following term $\sum_{i=1}^n (x_i^2 - x_iy_i)$ over $(x_1,\ldots,x_n)$, $(y_1,\ldots,y_n)$ non-negative such that $\sum_{i=1}^n x_i ... 0answers 165 views ### Convexity of a Set Consider the following function, $$f(x, y) = e^{m e^{-y}+n e^{-x}-x-y} \left(a x e^y+b e^x y+c x y\right)$$ where$a, b, c, m$and$n$are positive constants. I want to show$f(x, y)$is ... 1answer 192 views ### Supremum of vector dot-product I am curious what is the precise math reasoning behind this: $$\sup \{ a_i^T u | \lVert u\rVert_2 \leq r \} = r \lVert a_i \rVert_2$$ It is on page 148, last line, of Boyd's Convex Optimization ... 2answers 345 views ### Proving quadratic function is bounded (vector input) I am reading Boyd's Convex Optimization textbook and I am looking at example 3.22, line 2. It says$y^T x - \frac12 x^T Q x$is bounded from above for all possible values of$y\$. Also, it is ...
In Boyd's Convex Optimization text, on page 86, there is the following equation: $$f(x) = h(g(x)) = h(g_1(x), \cdots, g_k(x))$$ Then, in order to show convexity, we take the second derivative, ...